GoG Galaxy is telling me that it cannot connect and ends itself.

I see that others are having issues with Galaxy and functionality as well, but different issues. If somebody's got a fix for the ESET issue, would love to hear it.

Looks like they rolled out a dud with this one...

I know this sounds rather basic/simplistic and you may have already done so but have you added all Galaxy-related folders to your AV whitelist? Also, check to see if GalaxyClientService and GalaxyCommunication start up at any point when you try to launch Galaxy.

Check your firewall logs and see what you're blocking.

The game time tracking article mentions 3 executables that need Internet access for that feature, though there may be others for other features. For example, for achievements, the game executable also needs Internet access. You may want to whitelist everything under these 2 folders and call it a day:
C:\Program Files (x86)\GOG Galaxy\
C:\ProgramData\GOG.COM\Galaxy\